nLab
Adams conjecture

Context

Bundles

Stable Homotopy theory

Contents

Idea

The Adams conjecture is a statement about triviality of spherical fibrations associated to certain formal differences of vector bundles (K-theory classes) via the J-homomorphism. The conjecture was stated in (Adams 63, conjecture 1.2), for vector bundles of rank up to two over a finite CW-complex, which was proven in (Adams 63, theorem 1.4). A general proof was then given in (Quillen 71).

The Adams conjecture/Adams-Quillen theorem serves a central role in the identification of the image of the J-homomorphism.

Statements

Let XX be (the homotopy type of) a topological space. For V:XBOV \;\colon\; X \longrightarrow B O classifying a real vector bundle on XX, the corresponding spherical fibration is classified by the composite

J(V):XVBOJBGL 1(𝕊) J(V) \;\colon\; X \stackrel{V}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S})

with the delooped J-homomorphism. This descends to a map from topological K-theory to spherical fibrations.

Now for LL a line bundle on some XX and for non-vanishing kk \in \mathbb{Z}, John Adams observed that the spherical fibration associated with the difference L kLKO(X)L^{\otimes k} - L \in K O(X) has the property that some kk-fold multiple of it has trivial spherical fibration, hence that there is NN \in \mathbb{N} for which

J( k N(L kL))=0. J\left( \oplus^{k^N} (L^{\otimes k} - L) \right) = 0 \,.

Noticing that LL k=Ψ k(L)L \mapsto L^{\otimes^k} = \Psi^k(L) is the kkth Adams operation on K-theory applied to the line bundle LL, John Adams then conjectured that the above is true for all vector bundles VV in the form

J( k N(Ψ k(V)V))=0. J\left( \oplus^{k^N} (\Psi^k(V) - V) \right) = 0 \,.

References

General

The conjecture originates in:

Textbook accounts:

Review:

The proof of the Adams conjecture is originally due to

The proof using algebraic geometry is due to

Yet another proof via Becker-Gottlieb transfer is due to

In equivariant cohomology

The generalization to equivariant cohomology (equivariant K-theory) is discussed in